Behaviour of mixed finite element and multipoint ...

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XVI International Conference on Computational Methods in Water Resources (CMWR-XVI) Ingeniørhuset

Behaviour of mixed finite element and multipoint point flux approximations for flow with discontinuous coefficients
Author:Vincent FONTAINE <> (Institut de Mécanique des Fluides et des Solides)
Anis YOUNES <> (Institut de Mécanique des Fluides et des Solides)
Presenter:Vincent FONTAINE <> (Institut de Mécanique des Fluides et des Solides)
Date: 2006-06-18     Track: General Sessions     Session: General

We study the numerical behaviour and the relationships between some numerical methods derived from the Mixed Finite Element (MFE) formulation and from the Control Volume formulation using the Multi-Point Flux Approximation (MPFA) of Aavatsmark et al. (1996). All methods are locally mass conservative and are well suited to solve the flow problem for both steady state (elliptic case) an transient (parabolic case) on a general irregular grid with anisotropic and heterogeneous discontinuous coefficients. Hybridization of the standard MFE method allows to obtain a symmetric positive definite system which is not, in general, an M-matrix. Moreover, the mixed hybrid finite element method uses more unknowns (one per edge or face) than the finite volume formulation (one per element). On the other hand, for a general element grid, the MPFA method does not give a symmetric matrix of coefficients and monotonicity issues are known to arise for high aspect ratios combined with skewed of computational grids. In this work, we propose two variants of the MFE method: (i) the lumped-MFE method, based on the equivalence between MHFE method and the P1-nonconforming Galerkin method for the Laplace equation and (ii) the multipoint-MFE method based on Multipoint finite volume discretization using the framework of the modified mixed finite element space. We discuss connections between the different approaches and perform numerical experiments for both steady state and transient cases to compare (number of unknowns, matrix properties, condition number) and study the numerical behaviour of these methods (unphysical oscillations, CPU time).